From Zagreb to Trieste there is a travel distance of about 230 km, which corresponds to a bus travel time of about 3.5 to 4 hours. Trough out the year there are several daily bus departures between Zagreb and Trieste.
If you are travelling to the Trieste airport, be aware that the airport is situated about 40 km from the city centre in Trieste. From where the bus stops you although have an frequent airport bus going to the airport, you also have easy access to the Italian train network.
Here you can see bus timetable for buses between Zagreb and Trieste: (Bus schedule for the route Trieste to Zagreb you can find here)